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17.6 Find a maximal ideal in b) Z12;

17.6 Find a maximal ideal in b) Z12;

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Topic: Maximal Ideals in Ring Theory **Problem Statement:** **17.6** Find a maximal ideal in **b) \( \mathbb{Z}_{12} \);** **Explanation:** - The problem asks us to identify a maximal ideal within the ring \( \mathbb{Z}_{12} \), which represents the integers modulo 12. - **Maximal Ideal:** An ideal \( I \) in a ring \( R \) is called maximal if there are no other ideals contained between \( I \) and \( R \) except for \( I \) itself and \( R \). **Steps to Solve:** 1. **Identify Ideals in \( \mathbb{Z}_{12} \):** - The ideals in \( \mathbb{Z}_{12} \) correspond to the divisors of 12 because \( \mathbb{Z}_{n} \) is a principal ideal ring. - These are \( (0), (2), (3), (4), (6), (12) \). 2. **Determine Maximal Ideals:** - An ideal \( (d) \) in \( \mathbb{Z}_{n} \) is maximal if and only if \( n/d \) is a prime number. - Calculate \( n/d \) for each ideal: - \( 12/2 = 6 \) - \( 12/3 = 4 \) - \( 12/4 = 3 \) - \( 12/6 = 2 \) - Only \( 12/4 = 3 \) and \( 12/6 = 2 \) are prime numbers. 3. **Conclusion:** - The maximal ideals in \( \mathbb{Z}_{12} \) are \( (4) \) and \( (6) \). This approach explains how one determines maximal ideals in \( \mathbb{Z}_{12} \), highlighting both the theoretical understanding and practical calculation.
Transcribed Image Text:### Topic: Maximal Ideals in Ring Theory **Problem Statement:** **17.6** Find a maximal ideal in **b) \( \mathbb{Z}_{12} \);** **Explanation:** - The problem asks us to identify a maximal ideal within the ring \( \mathbb{Z}_{12} \), which represents the integers modulo 12. - **Maximal Ideal:** An ideal \( I \) in a ring \( R \) is called maximal if there are no other ideals contained between \( I \) and \( R \) except for \( I \) itself and \( R \). **Steps to Solve:** 1. **Identify Ideals in \( \mathbb{Z}_{12} \):** - The ideals in \( \mathbb{Z}_{12} \) correspond to the divisors of 12 because \( \mathbb{Z}_{n} \) is a principal ideal ring. - These are \( (0), (2), (3), (4), (6), (12) \). 2. **Determine Maximal Ideals:** - An ideal \( (d) \) in \( \mathbb{Z}_{n} \) is maximal if and only if \( n/d \) is a prime number. - Calculate \( n/d \) for each ideal: - \( 12/2 = 6 \) - \( 12/3 = 4 \) - \( 12/4 = 3 \) - \( 12/6 = 2 \) - Only \( 12/4 = 3 \) and \( 12/6 = 2 \) are prime numbers. 3. **Conclusion:** - The maximal ideals in \( \mathbb{Z}_{12} \) are \( (4) \) and \( (6) \). This approach explains how one determines maximal ideals in \( \mathbb{Z}_{12} \), highlighting both the theoretical understanding and practical calculation.
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